I am trying to learn "Geometric invariant theory" like it was introduced by Mumford. But I do not have a strong background in algebraic geometry since I work in geometric topology and geometry.

So my question is if there is any nice reference where people explain geometric invariant theory from a geometric viewpoint. In particular, I am looking for a good reference where the analogies between algebraic geometry and differential geometry are pointed out.

I think these kinds of questions are a bit silly; imagine asking about a reference for elliptic operators that doesn't use any theory of Sobolev spaces! At the very least, I think the OP should clarify more carefully what is his background in (presumably complex) differential geometry, and precisely how far along he is in learning algebraic geometry.
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Jason StarrAug 2 '12 at 18:29

4 Answers
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If you just want to get a feeling for invariant theory, here are some books that aren't necessarily comprehensive but nevertheless are enlightening at a more leisurely pace as compared to GIT, which would be useful for someone who isn't as familiar with algebraic groups and algebraic geometry:

Santos and Rittatore - Actions and Invariants of Algebraic Groups: Minimal prerequisites. A very gentle introduction to some aspects of invariant theory, including some motivation via Hilbert's 14th problem. This book also contains most of the required theory of linear algebraic groups.

Dolgachev - Lectures on Invariant Theory: This takes a more geometric viewpoint and might be something you are interested in. This only requires some basic knowledge of algebraic geometry.

Schmitt - Geometric Invariant Theory and Decorated Principal Bundles: this might also be interesting if you are interested in the geometric applications and the related geometry, though I haven't looked into this book very much, but Part 1 does contain a fairly leisurely-looking introduction to GIT

There is also Popov's and Vinberg's treatise "Invariant Theory" in the Ecyclopedia of Mathematical Sciences Volume 55 (Springer) which contains a good summary of the classical results in characteristic zero.

I would recommend a look at chapter 8 of the third edition of Geometric invariant theory by Mumford, Forgarty and Kirwan. It describes a connection between GIT and Hamiltonian group actions in symplectic geometry.